The conjecture of Sally on the Hilbert function for curve singularities
Let \(A\) be a one-dimensional Cohen-Macaulay local ring with maximal ideal \({\mathfrak m}\). \textit{J. Sally} [in ``Numbers of generators of ideals in local rings'', Lect. Notes Pure Appl. Math. 35 (1978; Zbl 0395.13010)] conjectured that if \(A\) is of small embedding dimension, then the Hilbert function length\(({\mathfrak m}^ n/{\mathfrak m}^{n + 1})\) is nondecreasing. \textit{F. Orecchia} [Manuscr. Math. 32, 391-405 (1980; Zbl 0445.13008)] proved that there exist reduced one-dimensional local rings of embedding dimension \(\geq 5\) with decreasing Hilbert function. In this paper, the conjecture is settled in the affirmative for equicharacteristic rings of embedding dimension 3. The key point in the proof is the fact that \(A\) is defined by a codimension 2 ideal so that one can use the Hilbert-Burch theorem.
scientific article; zbMATH DE number 436479